iPad mini has equal performance (iOS code is more efficient than Android), great screen (but not HD).

iPad mini is better in terms of perfect build materials and quality, superb battery life, future 1st and 3rd party support. iPad Mini has a lot more Apps and Apps that are optimized and are simply better. iPad mini is 4:3 shape so its more comfortable to use and things look better like Browser and some Apps.

The Nexus 7 is better in terms of price (cheaper), software (Android is much better and advanced than iOS), customization (make it yours) and generally less restricted. Unlike the iPad, Nexus 7 can be held with one hand (XL Hand people only). Nexus 7 shows videos much better and a select few Apps (like Doodle Jumps long vertical requirements). The screen is 720p HD which is awesome.

What should you get?
It depends on you. iPad mini is a better gift and better tablet (hardware and Apps) and is stupidly simple. The Nexus 7 has more potential, and if you have special needs or/and can get your hands dirty its a better device AND cheaper.